Covered in lectures. Check back once the chapter is concluded.
4 Holomorphic functions
The following definition is a key concept for this course.
Definition 4.1 A complex function \(f\colon U\to\C\) with domain an open set \(U\subset \C\) is complex differentiable at a point \(z_0\in U\) if the limit
\[f'(z_0)=\frac{\6f}{\6z}(z_0)=\lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0} \tag{4.1}\]
exists. We call \(f\) holomorphic on \(U\) if \(f\) is complex differentiable at every \(z_0\in U.\) The set of all holomorphic functions on \(U\) is denoted \(\cO(U).\) A function that is holomorphic1 on \(U=\C\) is entire.
Proposition 4.1 Let \(f, g\colon U\to\C\) be complex differentiable at \(z_0.\) Then:
- \(f+g\) is complex differentiable at \(z_0\) with \[(f+g)'(z_0)=f'(z_0)+g'(z_0).\]
- \(fg\) is complex differentiable at \(z_0\) with\[(fg)'(z_0)=f'(z_0)g(z_0)+f(z_0)g'(z_0).\]
- If \(g(z_0)\neq0,\) then \(f/g\) is complex differentiable at \(z_0\) with\[(f/g)'(z_0)=\frac{f'(z_0)g(z_0)-f(z_0)g'(z_0)}{g(z_0)^2}.\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Example 4.1
Covered in lectures. Check back once the chapter is concluded.
Proposition 4.2 Let \(f\colon U\to\C,\) \(g\colon V\to\C\) be complex functions with \(f(U)\subset V,\) and \(U,\) \(V\) open. Suppose that \(f\) is complex differentiable at \(z_0\in U\) and that \(g\) is complex differentiable at \(w_0=f(z_0)\in V.\)
Then \(g\circ f\) is complex differentiable at \(z_0\) and\[(g\circ f)'(z_0)=g'(w_0)f'(z_0). \tag{4.2}\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Covered in lectures. Check back once the chapter is concluded.
Proposition 4.3 For a complex function \(f=u+iv\) the following are equivalent:
- \(f\) is complex differentiable at \(z_0\)
- \(f\) is real differentiable at \(z_0\) and the Cauchy–Riemann equations \[\frac{\6u}{\6x}(z_0) = \frac{\6v}{\6y}(z_0), \qquad \frac{\6u}{\6y}(z_0) = -\frac{\6v}{\6x}(z_0), \tag{4.3}\] hold.
In this case, \(f'(z_0)=\frac{\6u}{\6x}+i\frac{\6v}{\6x}.\)
Proof.
Covered in lectures. Check back once the chapter is concluded.
Example 4.2
Covered in lectures. Check back once the chapter is concluded.
Example 4.3
Covered in lectures. Check back once the chapter is concluded.
Example 4.4
Covered in lectures. Check back once the chapter is concluded.
Example 4.5
Covered in lectures. Check back once the chapter is concluded.
Theorem 4.1 Let \(f\colon U\to \C\) be a holomorphic function and \(z_0\in U\) with \(f'(z_0)\neq0.\) Then there exist open sets \(V, W\subset\C\) with \(z_0\in V\subset U\) and \(w_0=f(z_0)\in W\) with the property that the restriction \(f|_V\) becomes a bijection \(V\to W\) with holomorphic inverse function \((f|_V)^{-1}\colon W\to V\) and \[\frac{\6(f|_V)^{-1}}{\6w}(w)=\frac{1}{f'\bigl((f|_V)^{-1}(w)\bigr)}. \tag{4.4}\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Remark 4.1. Our proof is a simple application of the ordinary inverse function theorem. We will see two different proofs later in the course.
Example 4.6
Covered in lectures. Check back once the chapter is concluded.
Questions for further discussion
- \(x\mapsto x^3\) is a real differentiable bijection whose inverse function is not differentiable at \(x=0.\) What about the complex analogue \(z\mapsto z^{1/3}\)?
4.1 Exercises
Exercise 4.1
Write the following complex functions in the form \(f=u+iv\): \[f_1(z)=\sin(z),\qquad f_2(z)=e^{z^2},\qquad f_3(z)=\cosh(z)\]
Exercise 4.2
At which points \(z\in\C\) are the following functions complex differentiable? At which points are they real differentiable? \[\begin{align*} f_1(z)&=z,&f_2(z)&=\ol{z}, &f_3(z)&=z^3+z,\\ f_4(z)&=\frac{1}{2iz}, &f_5(z)&=|z|^2, &f_6(z)&=\frac{|z|^2}{\ol{z}} \end{align*}\]
Exercise 4.3
Let \(f\colon V\to W\) be a bijection of open sets \(V, W\subset\C.\) Assume that \(f\) is complex differentiable at \(z_0\) and that \(f^{-1}\) is complex differentiable at \(w_0=f(z_0).\) Show that \(f'(z_0)\neq0.\) (The same result holds for real differentiable maps to show \(\det J_f(z_0)\neq0\).)
Exercise 4.4
Let \(f\colon U\to\C\) be a holomorphic function with domain an open set \(U\subset\C.\) Suppose also that \(f'(z)\) is holomorphic. Write \(f(z)=u(x,y)+iv(x,y),\) where \(z=x+iy.\) As we will see, a consequence of \(f\) being holomorphic, is that \(u\) and \(v\) have continuous second order derivatives. Show that:
- \(|f'(z)|^2=\Bigl(\frac{\6u}{\6x}\Bigr)^2+\Bigl(\frac{\6v}{\6x}\Bigr)^2=\Bigl(\frac{\6u}{\6y}\Bigr)^2+\Bigl(\frac{\6v}{\6y}\Bigr)^2\)
- Both \(u,\) \(v\) satisfy the Laplace equation \(\Delta(u)=\Delta(v)=0,\) where the Laplace operator is defined as \(\De=\frac{\6^2}{\6 x^2}+\frac{\6^2}{\6 y^2}.\)
- Fix \(n\in\N.\) Use b. to find a solution to \(\De(u)=0\) that satisfies \(u(z)=\cos(n\th)\) for all \(|z|=1,\) \(z=e^{i\th}\) on the unit circle.
- Does the converse to b. hold?
Exercise 4.5
Find subdomains where \(\sin,\cos,\sinh,\cosh,\tan,\tanh\) are injective. Compute the derivative of an inverse using the chain rule. Deduce an explicit formula for the derivative of an inverse.
Exercise 4.6
Show that the set \(\cO(U)\) of holomorphic functions on \(U\) becomes a ring under the operations of point-wise addition and multiplication: \[\begin{align*} (f+g)(z)&=f(z)+g(z), &(fg)(z)&=f(z)g(z). \end{align*}\]
Exercise 4.7
Let \(f\colon\C\to\C\) be an entire function with image \(f(\C)\subset\R.\) Prove that \(f\) is constant.
Exercise 4.8
Let \(U\) be an open set and assume that \(U\) is path-connected, meaning that for all \(z_0, z_1\in U\) there exists a continuously differentiable map \(\ga\colon[0,1]\to U\) such that \(\ga(0)=z_0,\) \(\ga(1)=z_1.\)
Let \(f\colon U\to\C\) be a holomorphic function such that \(f'(z)=0\) for all \(z\in U.\) Prove that \(f(z)\) is a constant function.
Hint: Consider the real and imaginary parts of \(f\circ\ga.\)
Show that the only entire functions \(f(z)\) satisfying \(f'(z)=C f(z)\) for a constant \(C\in\C\) are the functions \(f(z)=De^{Cz},\) \(D\in\C.\)
Find all entire solutions \(f(z)\) of \(f''(z)=f(z).\)
Hint: Consider \(g=f+f',\) \(h=f-f'\)
From Greek holos ‘whole, complete’ and morphē `form, shape↩︎